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Review of Applications will begin on February 13,
2012. Award notifications will be sent by May 1, 2011.

All applicants must send transcripts (unofficial is fine) and two letters of recommendation from professors familiar with the applicant's abilities to the program director (see right side).

Eligibility

You must be a U.S. citizen or U.S. permanent resident to be considered.
You must be an undergraduate student during the program running dates.
Students from underrepresented groups, especially first generation college students and American Indians, are encouraged to apply.

Program Director:
Dr. Jennifer McLoud-Mann
The University of Texas at Tyler
3900 University Blvd
RBN 4008
Tyler, TX 75799
(903) 566-7327
jmcloud@uttyler.edu

Housing Costs

Students will be required to stay in the Patriot Village Apartments at a rate of $15 per day ($840 for the entire 8 weeks).

2012 Program Dates

June 18 - August 10
Students must arrive to campus by June 17.

Program Funding

It is anticipated that this program will be funded for the summers of 2011, 2012, and 2013. All awards made are contingent on funds from the NSF.

What do students do in the UT Tyler Mathematics REU?

Students will work on research in groups of three students that are each led by a faculty advisor. The research groups will be centered around three exciting areas of mathematical research.

Program Benefits:

Work closely with faculty in gaining valuable experience in the process of mathematical research. See research topics below.
A $4000 stipend to pay for housing, food, and entertainment.
Learn how to write and typeset a professional mathematics article using LaTeX.
Learn how to prepare and deliver an excellent talk on mathematics. Students will be funded to attend national conferences to present the results of their research experience.
Get information on the process of choosing and applying for graduate programs in mathematics.
Fun team-building social activities, such as
- Outdoor activities (Camp Tyler ropes course, canoeing at Lake Tyler)
- A trip to Austin to visit museums and other fun locations.
- A trip to Dallas to visit the aquarium
Students will have full access to The University of Texas at Tyler's facilities, including the Patriot Center (swimming pool, gym) and the library.

Summer 2012 Research Topics:

Knot Theory -- led by Dr. Jennifer McLoud-Mann

Knot theory is very accessible to undergraduate students, and Dr. McLoud has successfully mentored 5 undergraduate research projects in this area. Student authors have presented their work at regional/state meetings, and almost all have published or submitted manuscripts for publication. Several problems related to Dr. McLoud’s research interests remain open for investigation:

Cell Knots. Given a lattice L, a lattice knot K is a closed nonintersecting polygonal path
that consists of unit edges whose endpoints lie on the lattice. The minimum edge number of a knot K is the smallest number of unit sticks that can be used to form the knot K, denoted mK. Minimum edge numbers for trefoil knots have been found in many standard lattices. A natural Voronoi tessellation of space is associated with a given lattice, and the 3D Voronoi cells that tessellate can be used to construct a cell knot. The minimum number of cells to form a knot K is called the minimum cell number of K and is denoted MK. The connection between mK and MK for a given knot K can be studied. For the simple cubic lattice, MK = 2·mk. Preliminary results indicate that the same result holds for the simple hexagonal lattice. Exploring the connection between mK and MK in the face-centered cubic lattice and the body-centered cubic lattice, important to chemists, will be interesting and valuable.
Lattice Stick Number. For a given lattice knot K, instead of focusing on the minimum
number of edges, one can focus on the minimum number of sticks to create it (called lattice stick number). A stick is a maximum line segment in the knot consisting of one or many unit edges. Stick numbers in the simple cubic lattice and simple hexagonal lattice have been studied. Similar work in the face-centered and body-centered cubic lattices remains unexplored and seems feasible.
Composite Knots. Given two knots K and L, an operation can be performed to construct
the composite knot of K and L, denoted K#L. For a specific invariant such as the minimal
edge number, determining the relationship between mK, mL, and mK#L would be interesting. In the simple cubic lattice, the edge number of the composition of two knots can always be reduced by two. That is, mK#L ≤ mK + mL − 2. Students can explore the relationship between these minimal edge numbers in other lattices, such as the simple hexagonal lattice, face-centered cubic lattice, and the body-centered cubic lattice.

Tiling Theory -- led by Dr. Casey Mann

Students will be given the opportunity to pursue several possible lines of inquiry, including:

Convex tiles

All triangles and quadrilaterals are known tile the plane, and convex hexagons that tile the plane have been classified; however, a complete classification of convex pentagons that tile the plane has not been found. It is also known that for n ≥ 7, convex n-gons do not tile the plane. One question, then, is if there is a maximum finite Heesch number for convex polygons. Another question involves simply identifying some examples of convex polygons that have a finite Heesch number greater than 0. To date, only a few examples of convex polygons with Heesch number 1 have been identified. This problem lends itself to computerized searches that would introduce the students to large-scale computations on a computer cluster. Dr. Mann has developed an algorithm than can be used to discover regular polygons (up to some predetermined number of sides and a predetermined maximum vertex valence) that have Heesch number at least 1. Those tiles that have Heesch number at least one are then candidates as tiles that either tile
the plane or have Heesch number 1 or greater; the former possibility may result in the discovery of convex pentagons that tile the plane, which may result in the discovery of new interesting nontilers with Heesch number 1 or greater. REU students would implement the algorithm on the computing cluster and run it to get results.

Generalized Matching Rules

Very recently (DeWeese, Corinaldi - 2010), a hexagon whose edges are marked with combinatorial matching rules was found to have Heesch number 11. The students in my summer 2011 REU research group wrote computer algorithms that can enumerate and find the Heesch numbers and other tiling properties of such tiles. In summer 2012, students could use these computer algorithms to explore questions like:

Find all "Conway-like" tiling criteria for polyforms with generalized edge matching rules.
Explore tiling properties of 2- and 3-polyominoes with generalized edge matchng rules.
Develop generalized vertex matching rules and develop further computer algorithms to explore the properties of tiles so marked.

Transitivity Classes in Tiling Symmetry Groups.

Tiling theorists are interested in algebraic properties of tilings that are forced by the tile or tiles that comprise the tiling (e.g., the idea of aperiodic tiles mentioned above: nonperiodicity is a property of the symmetry group of the tiling). Much work has been done in the area of tiles that admit only tilings in which the number of transitivity classes of tiles with respect to the symmetry group of the tilings is exactly some integer N; such tiles are called N-isohedral. Students could examine the problem of finding equilateral tiles which admit only edge-to-edge tilings in which the number of transitivity classes of edges is exactly N; such tiles would be called N-isotoxal. Variations on this theme involving the vertices of tilings or the flags of tilings could be explored as well.

Combinatorics on Words -- led by Dr. David Milan

Students working in this topic will be introduced to the subject of infinite square-free words and nonrepetitive vertex colorings of graphs. A word is said to be square-free if it has no subword that is repeated twice in a row. In 1906, Thue discovered an infinite square-free word on three generators. One related and highly active area is nonrepetitive vertex colorings of graphs. In this context, a vertex coloring is an assignment of a letter to each vertex. A simple path in a colored graph can then be thought of as a word. A vertex coloring is nonrepetitive, if all such words are square-free.

The minimum number of colors required for a nonrepetitive coloring of a graph G is called the Thue number π(G). For example, Thue’s result implies that π(P) = 3 for any path P. One stunning result gives the upper bound π(G) ≤ 16d² where d is the maximum degree of a vertex in G.

Students will consider some of the following questions:

Square-Free Words
Many results about nonrepetitive vertex colorings make use of square-free words with added properties. REU participants will consider questions such as:

For which values k do there exist infinite square-free words having no repeated letters over gaps of length k?
Is there a square-free word on 3 letters that avoids palindromes of length at least 5?

These questions could be solved in many ways. Students experienced in computer programming could write code that will help search for solutions. There are also combinatorial approaches similar to the method Thue used to find his results.

Graph Colorings
In addition to finding words with interesting combinatorial properties, students will look for applications to graph coloring problems. The proof that the Thue number of a tree is 4 led to a result called the leveling lemma. Answers to the above questions could be used in a similar way, suggesting the related questions:

Given square-free words on 3 letters that avoid long palindromes, what is a suitable generalization of the leveling lemma?
Given a new leveling lemma, what progress can be made on the question in about whether every graph admits a subdivision with Thue number 3?

Research Experience for Undergraduates in Mathematics